We present mathematical models to study the mechanism of interaction of tumour infiltrating cytotoxic lymphocytes (TICLs) with tumour cells. We focus on the phase spaces of the systems and the nature of the solutions for the cell densities in the short and long term. The first model describes the production of offspring through cell proliferation, death and local kinetic interactions. The second model characterises the spatial distribution dynamics of the cell densities through reaction diffusion, which describes the random movement of the cells, and chemotaxis, which describes the immune cell movements towards the tumour cells. We then extend these models further to incorporate the effects of immunotherapy by developing two new models. In both situations, we analyse the phase spaces of the homogeneous models, investigate the presence of travelling wave solutions in our systems, and provide numerical simulations. Our analysis shows that cancer dormancy can be attributed to TICLs. Our study also shows that TICLs reduce the tumour cell density to a cancer dormant state but even with immunotherapy do not completely eliminate tumour cells from body tissue. Travelling wave solutions were confirmed to exist in the heterogeneous model, a linear stability analysis of the homogeneous models and numerical simulations show the existence of a stable tumour dormant state and a phase space analysis confirms that there are no limit cycles. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2013.

We conduct a comprehensive investigative review of solution generating algorithms for the Einstein field equations governing the gravitational behaviour of an isolated neutral static spherical distribution of perfect fluid matter. Traditionally, the master field equation generated from the condition of pressure isotropy has been interpreted as a second order ordinary differential equation. However, since the pioneering work of Wyman (1949) it was observed that more success can be enjoyed by regarding the equation as a first order linear differential equation. There was a resurgence of the ideas of Wyman in 2000 and various researchers have been able to generate complete solutions to the field equations up to certain integrations. These have been accomplished by working in Schwarzschild (curvature) coordinates, isotropic coordinates, area coordinates and a coordinate system written in terms of the redshift parameter. We have utilised Durgapal–Banerjee (1983) coordinates and produced a new algorithm. The algorithm is used to generate new classes of perfect fluid solutions as well as to regain familiar particular solutions reported in the literature. We find that our solution is well behaved according to elementary physical requirements. The pressure vanishes for a certain radius and this establishes the boundary of the distribution. Additionally the pressure and energy density are both positive inside the radius. The energy conditions are shown to be satisfied and it is particularly pleasing to have the causality criterion satisfied to ensure that the speed of light is not exceeded by the speed of sound. We also report some new solutions using the algorithms proposed by Lake (2006). / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

In this thesis we study spherically symmetric spacetimes related to the Einstein field equations. We consider only neutral matter and apply the Einstein field equations with isotropic pressures. Our object is to model relativistic stellar systems. We express the Einstein field equations and the condition of pressure isotropy in terms of Schwarzschild coordinates and isotropic coordinates. For Schwarzschild coordinates we consider the transformations due to Buchdahl (1959), Durgapal and Bannerji (1983), Fodor (2000) and Tewari and Pant (2010). The condition of pressure isotropy is integrated and new exact solutions of the field equations are obtained utilizing the transformations of Buchdahl (1959) and Tewari and Pant (2010). These exact solutions are given in terms of elementary functions. For isotropic coordinates we can express the condition of pressure isotropy as a Riccati equation or a linear equation. An algorithm is developed that produces a new solution if a particular solution is known. The transformations reduce to a nonlinear Bernoulli equation in most instances. There are fundamentally three new classes of solutions to the condition of pressure isotropy. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

The study of embeddings is applicable and signicant to higher dimensional theories of our universe, high-energy physics and classical general relativity. In this thesis we investigate local and global isometric embeddings of four-dimensional spherically symmetric spacetimes into five-dimensional Einstein manifolds. Theorems have been established that guarantee the existence of such embeddings. However, most known explicit results concern embedded spaces with relatively simple Ricci curvature. We consider the four-dimensional gravitational field of a global monopole, a simple non-vacuum space with a more complicated Ricci tensor, which is of theoretical interest in its own right, and occurs as a limit in Einstein-Gauss-Bonnet Kaluza-Klein black holes, and we obtain an exact solution for its embedding into Minkowski space. Our local embedding space can be used to construct global embedding spaces, including a globally at space and several types of cosmic strings. We present an analysis of the result and comment on its signicance in the context of induced matter theory and the Einstein-Gauss-Bonnet gravity scenario where it can be viewed as a local embedding into a Kaluza-Klein black hole. Difficulties in solving the five-dimensional equations for given four-dimensional spaces motivate us to investigate which embedded spaces admit bulks of a specific type. We show that the general Schwarzschild-de Sitter spacetime and the Einstein Universe are the only spherically symmetric spacetimes that can be embedded into an Einstein space with a particular metric form, and we discuss their five-dimensional solutions. Furthermore, we determine that the only spherically symmetric spacetime in retarded time coordinates that can be embedded into a particular Einstein bulk is the general Vaidya-de Sitter solution with constant mass. These analyses help to provide insight to the general embedding problem. We also consider the conformal Killing geometry of a five-dimensional Einstein space that embeds a static spherically symmetric spacetime, and we show how the Killing geometry of the embedded space is inherited by its bulk. The study of embedding properties such as these enables a deeper mathematical understanding of higher dimensional cosmological models and is also of physical interest as conformal symmetries encode conservation laws. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2012.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2008. / In this paper, tracking tools are developed based on object features to robustly track the object using particle filtering. Automatic on-line initialisation techniques use motion detection and dynamic background modelling to extract features of moving objects. Automatically adapting the feature models during tracking is implemented and tested.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--Stellenbosch University, 2008. / The NLS is a universal equation of the class of nonlinear integrable systems. The aim of this thesis is to study the NLS numerically. More speci cally, an algorithm is developed to calculate its nonlinear spectrum. The nonlinear spectrum is then used as a diagnostic for numerical studies of the NLS. The spectrum consists of a discrete part, further subdivided into the main part, the auxiliary part, and the continuous spectrum. Two algorithms are developed for calculating the main spectrum. One is based on Floquet theory, rst implemented by Overman [12]. The other is a direct calculation of the eigenvalues by Herbst and Weideman [16]. These algorithms are combined through the marching squares algorithm to calculate the continuous spectrum. All ideas are illustrated by numerical examples.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2007. / The face is one of the most characteristic parts of the human body and has been used by people for personal identification for centuries. In this thesis an automatic process for frontal face recognition from 2–dimensional images is presented based on principal component analysis. The goal is to use these concepts in eventual face–recognizing login software. The first step is detecting faces in images that are allowed a certain degree of clutter. This is achieved by skin colour detection in the HSV colourspace. This process indicates the area of the image most likely corresponding to the face. Extracting the face is achieved by morphological processing of this area of the image. The face is then normalized by a transformation that uses the eye coordinates as input. Automatic eye detection is implemented based on colour analysis of the facial images and a 91.1% success rate is achieved. Recognition of the normalized faces is achieved using eigenfaces. To calculate these, a large enough database of facial images is needed. The xm2vts database is used in this thesis as the images have very constant lighting conditions throughout – an important factor affecting the accuracy of the recognition stage. Distinction is also made between identification and verification of faces. For identification, up to 80.1% accuracy is achieved, while for verification, the equal error rate is approximately 3.5%.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2009. / In this thesis, we investigate two numerical methods to price nancial options. We look at two types of options, namely European options and Asian options. The numerical methods we use are the nite di erence method and numerical inversion of the Laplace transform. We apply nite di erence methods to partial di erential equations with both uniform and non-uniform spatial grids. The Laplace inversion method we use is due to Talbot. It is based on the midpoint-type approximation of the Bromwich integral on a deformed contour. When applied to Asian options, we have the problem of computing the hypergeometric function of the rst kind. We propose a new method for numerically calculating the hypergeometric function. This method too is based on using Talbot contours. Throughout the thesis, we use the Black-Scholes equation as our benchmark problem.

Thesis (MSc (Applied Mathematics))--Stellenbosch University, 2008. / The Korteweg-de Vries (KdV) hierarchy is an important class of nonlinear evolution equa- tions with various applications in the physical sciences and in engineering. In this thesis analytical solution methods were used to ¯nd exact solutions of the third and ¯fth order KdV equations, and numerical methods were used to compute numerical solutions of these equations. Analytical methods used include the Fan sub-equation method for constructing exact trav- eling wave solutions, and the simpli¯ed Hirota method for constructing exact N-soliton solutions. Some well known cases were considered. The Fourier spectral method and the ¯nite di®erence method with Runge-Kutta time dis- cretisation were employed to solve the third and the ¯fth order KdV equations with periodic boundary conditions. The one soliton and the two soliton solutions were used as initial conditions. The numerical solutions are obtained and compared with the exact solutions. The propagation of a single soliton as well as the interaction of double soliton solutions is modeled well by both numerical methods, although the Fourier spectral method performs better. The stability, consistency and convergence of these numerical methods were investigated. Error propagation is studied. The theoretically predicted quadratic convergence of the ¯nite di®erence method as well as the exponential convergence of the Fourier spectral method is con¯rmed in numerical experiments.

Thesis (MScEng (Mathematical Sciences. Applied Mathematics))--Stellenbosch University, 2008. / When building or improving a petrochemical plant, drawings are used extensively in the design process. However, existing petrochemical plants seldom match their drawings, or the drawings are lost, forcing the need to generate a 3D model of the structure of the plant. In this thesis photogrammetry is investigated as a method of generating a digital 3D model of an existing plant. Camera modeling, target extraction and 3D reconstruction are discussed in detail, and a real-world system is investigated.